Topographic mapping of the Quantum Hall Liquid using a Few-Electron Bubble

Our scanning probe technique was used to obtain a high-resolution map of the random electrostatic potential inside the quantum Hall liquid. A sharp metal tip, scanned above a semiconductor surface, sensed charges in an embedded two-dimensional electron gas. Under quantum Hall effect conditions, applying a positive voltage to the tip locally enhanced the 2D electron density and created a "bubble" of electrons in an otherwise unoccupied Landau level. As the tip scanned along the sample surface, the bubble followed underneath. The tip sensed the motions of single electrons entering or leaving the bubble in response to changes in the local 2D electrostatic potential.

Magnetic field evolution of a 4x4 microns SCA image. Magnetic field ranges from 6.35 Tesla (filling factor ν 1 in the bulk) with a step of 0.05 Tesla between successive frames. The tip voltage is fixed at +1V with respect to the 2DEG. As a result, a bubble of enhanced electron density forms underneath the tip and is dragged inside the 2DEG as the tip is scanned above the sample surface. The bubble is separated from the bulk by an incompressible strip at filling factor ν = 1. Electrons may tunnel across the strip in response to changes the local electrostatic potential of the 2DEG. The signal, governed by Coulomb blockade across the strip, exhibits a series of maxima and minima that result in the observed contour lines that map the random electrostatic potential in the 2DEG. As the field decreases, the position of the strip changes and the contour lines eventually disappear when more than one spin-split Landau level is filled in the bulk (ν >1).

Zoom into a 2.5x2.5 microns area in the bottom left region of the previous images. The tip voltage is maintained at +1V with respect to the 2DEG. Magnetic field now ranges from 7.0 Tesla to 5.8 Tesla, with a step of 0.05 Tesla between successive SCA images.

Evolution of 2x2 microns SCA images with tip voltage bias. These images are taken at a different location from the ones presented earlier. The magnetic field is held constant at 6.5 Tesla and the tip bias is varied from +1.5V to +0.1V. The change in tip bias between successive frames is 0.2V. Each contour line marks the position in the 2DEG plane where the number of electrons in the bubble changes by one. As the tip voltage decreases, the influence of the tip on the 2DEG is reduced and the contours eventually disappear.